Numerical Integration *** 3/1/13 EC What’s Ahead • A Case Study on Numerical Differentiation: Velocity Gradient for Blood Flow • Finite Difference Formulas and Errors • Interpolation-Based Formulas and Errors • Richardson Extrapolation Technique • Finite Difference and Interpolation-based Formulas for Second Derivatives We have learnt the limits of sequences of numbers and functions, continuity of functions, limits of difference quotients (derivatives), and even integrals are limits of Riemann sums. I = Z b a f(x)dx … Z b a fn(x)dx where fn(x) = a0 +a1x+a2x2 +:::+anxn. Apply Newton’s rules of differentiation to basic functions. Differentiation and Integration, both operations involve limits for their determination. Created by T. Madas Created by T. Madas Question 1 Evaluate the following. Integration: With a flow rate of 2x, the tank volume increases by x 2. ��È��Jb��--�ű��W�.H�z;��š߁V[�up���٥|ߑ�$Np��P���. A�+�I�S�pF(��u,Q�@�5h��8�P�k�R���5(��Q�[�?��_ �����xƫN�t�R����X3�R������$�~**�+�YYC^����1��@}� Chapter 5: Numerical Integration and Differentiation PART I: Numerical Integration Newton-Cotes Integration Formulas The idea of Newton-Cotes formulas is to replace a complicated function or tabu-lated data with an approximating function that is easy to integrate. Numerical differentiation and interpolation Abstract Numerical integration and differentiation are some of the m ost frequently needed methods in computational physics. Differentiation and integration provide two possible methods for businesses to organize their operations and projects. (���%�R����z��x����G��x$��x�ˆ2�n�U�X�&Nrh �LK,�Ryᚣʃ �e�[+@��{f@D06�2 ��� k4R�k@�锵D�4ʁ�����cDx�A�Dr�\�w��~�ˆ8�����]�l�b�^f��y�zl��^�ٙ����٪&R�yU6՗p�a=��8�1)?e������A�O�_�6Y��w�{�]4ٺx���uN��gˢ\%���K���M�o~��v�͓�kWl�jG�…]8?�%���2��h�/�c[[�!�)/ڲ�G�� Differentiation and integration are basic mathematical operations with a wide range of applications in many areas of science. Integration is covered in tutorial 1. S�=��%���� �d�iF�y ��a�����w8�pj���[�;���b @,��[z�������̖�}�n��|�8eclÆ�'p��7� �E6�\>]�b��U�}Z߀����Z�v��;8�)W� Integration is covered in tutorial 1. d���uU�L�l*���2�±�U�غ�x%���&�VN���T|�' ���Pj��lŨ�؇��u$���K�XJ�bKPd�gө&���K�3�\7=/���8���� �~������K}Gj7qW�uG�#O�Хsgb 3�o�ڃl�]��z�]�"Q�we�.7����ٺ�hG%��&��i�w��i �q(�k�E�u���-)��]�d�̟��R�=���gGF{� After 3 minutes (x=3): the flow rate has reached 2x = 2×3 = 6 liters/min, h�b```� fN/a�13�0p,��(ʠ8���%GZ�~a�= ���&CÌL׀J�=������cg��;��a���}�6�`�h`�`�h`�h�`�h��h�� ��:l��4;s������|��'��� �e`S��Y��7f`s� �� p�5T ��t�"|"��.��d�� @�s�ZG< ���=���΂`�3����y�x��I�v��8I�,�� ާ�h"ëh"�ߢ,���Ƴ@�Y��߂h���c�0�#�(�\D��g�z����{L`dꞿo�O9�1�c�����qy|t�χdE���� +���c�P��T�,�,߇!H��Lj�s;[����,���=����m��0"�a��(�>��jD3�1�~����Q�=�U��$�Ah��i;5i6��w|ڻxDJ\4{�dq޲�ŕ).J�v�g�&� ��OЋ�f�8�ْ�}8�5C���*���eBb ,g������*>�櫂�\rj��,H�X�WH��f��r���p��-֬u�$�"D�^��>�-'�n�O�4��iX�^� �.{�����H{�U��p៝��. I = Z b a f(x)dx … Z b a fn(x)dx where fn(x) = a0 +a1x+a2x2 +:::+anxn. 2 • We have seen two applications: – signal smoothing – root finding • Today we look – differentation – integration The concept of understanding integrating a differential function gives the original function is very hard for a high school student. integration and integration to differentiation. On completion of this tutorial you should be able to do the following. That is, yex if and only if xy ln. &I����{�o�\}<>� Dr�y���y4�a�1��$��fWU��K4I�zw >�� On completion of this tutorial you should be able to do the following. Chapter 5: Numerical Integration and Differentiation PART I: Numerical Integration Newton-Cotes Integration Formulas The idea of Newton-Cotes formulas is to replace a complicated function or tabu-lated data with an approximating function that is easy to integrate. 0 Differentiation and Integration 1. 5.4 Exponential Functions: Differentiation and Integration Definition of the Natural Exponential Function – The inverse function of the natural logarithmic function f x xln is called the natural exponential function and is denoted by f x e 1 x. %%EOF Calculus is usually divided up into two parts, integration and differentiation. Integration formulas y D A B x C= + −sin ( ) A is amplitude B is the affect on the period (stretch or shrink) C is vertical shift (left/right) and D is horizontal shift (up/down) Limits: 0 0 sin sin 1 cos lim 1 lim 0 lim 0 x x x x x x The domain of {wp�ȴ���i�5z=�g!��<6{���@�K�M��0���Z�i� � +�4��՜��R�@�%~)�^"V����2��5�ƅ�萉��Ҡ�=ļ5��f��P�z�� endstream endobj 453 0 obj <>/Metadata 49 0 R/Pages 450 0 R/StructTreeRoot 68 0 R/Type/Catalog>> endobj 454 0 obj <>/MediaBox[0 0 595.4 842]/Parent 450 0 R/Resources<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI]>>/Rotate 0/StructParents 0/Tabs/S/Type/Page>> endobj 455 0 obj <>stream Math Methods Numerical Integration & Differentiation Project Rev 070105 A-2 Fig. 461 0 obj <>/Filter/FlateDecode/ID[<2E4C56239E86E34FBA67A222F313302C>]/Index[452 32]/Info 451 0 R/Length 60/Prev 411030/Root 453 0 R/Size 484/Type/XRef/W[1 2 1]>>stream Derivative: If the tank volume increases by x 2, then the flow rate must be 2x. Calculus is usually divided up into two parts, integration and differentiation. 483 0 obj <>stream Apply Newton’s rules of differentiation to basic functions. or an integral! DIFFERENTIATION . You proba-bly learnt the basic rules of differentiation and integration … Differentiation & Integration Formulas DIFFERENTIATION FORMULAS dx d (sin u) = cos u dx du (csc u) = −csc u cot u (cos u) = −sin u (sec u) = sec u tan u (tan u) = sec² u (cot u) = −csc² u (ln u) = 1⁄ u (e u) = eu (log a u) = 1⁄ u log a e INTEGRATION FORMULAS Note: a, b and c are constants; k is the integration … Result of calculations. It is therefore important to have good methods to compute and manipulate derivatives and integrals. �7��~�:�� g�h�b�h�����>Ž�c�����������Z/��9��ɡ���g{�G[vĝ��Gȃ�-oPB �"Yb�������)��$�` x��Y�n�F}7��#YT��% |M44�S�Z�-�6�ڒR��3�rI��S�x�9;{���4=y��ݔ�m����d�-��"�>=]o�뇿�W����ky{W�ۻu5���g�������ypus|Ă�� K✧���>�������?> Integration refers to how those components cooperate. Properties of the Natural Exponential Function: 1. Each is the reverse process of the other. A-3. h��Vmo�6�+���� (8ɼh; ɖ? p������ܤF����p�ָh\0�G˃�*:�5��8��g�8a����\����wk�t���F��A� ,�j/W7�ֱ(��Re�{o��W�-�ό^�%x?�I��=$`,Z���1,R���� �T����9{�B��T���p�6�&ʏ�Ըְ��N%K�@]�,a�(Y�vE��V]H9ʌ�όjJf��`��i_�q�9&-���[�_P��T��}�?׃2t_��(M�$%p�ب�֜G%Bw�D�^���� �@�,�^њ��S�x���Q���}B��V�x�n��b�s�%���,a��,��,�س,q*�g)��W�w�2�` D=� Explain differential coefficients. �fʐ��� �8YF����C0�j��� 7+J����P�L���8/+�f���=N�5�q7�w�p�ׁ�`c8 -��|pٷ;�m�)�̺��� �=�����:s:;��ř��nh�v����Q(Wt����2�Z9��{=U��LƽEH�*�c���@}fl$H��i�Z�3��J�&Hs�V��V|%�B���nW��{|o��xF�,��/������Y� This lesson basically explains when you send a function through the function generator it will show the differentiation function and Integrated function. Example: with the flow in liters per minute, and the tank starting at 0 . %PDF-1.5 %���� When performed over a full cycle and plot, the result is a sine wave like that shown in Fig. Interchange of Differentiation and Integration The theme of this course is about various limiting processes. Explain differential coefficients. h�bbd``b`� $k@��`��ׂ$�AbK���I �H02Y��&�3�0 � endstream endobj startxref The derivative of any function is unique but on the other hand, the integral of every function is not unique. endstream endobj 456 0 obj <>stream Quite often we are confronted with the need of evaluat-ing either the derivative f! Each is the reverse process of the other. Both differentiation and integration, as discussed are inverse processes of each other. Differentiation formulas . Two integrals of the same function may differ by a constant. 9}��7�n`YOrz�!�j����͞խ�2F��c��Ny3��ʂ����S���dP�)t��<1�T Differentiation refers to how a business separates itself into key components such as departments or product offerings. ("z��x�ඨ��c���z�h�G���/+�xEn� �MT�%"�zG��Mo����W%DX��Ƈ��. 452 0 obj <> endobj

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